fom
Posts:
1,677
Registered:
12/4/12
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Re: Matheology § 255
Posted:
Apr 20, 2013 1:06 PM
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On 4/20/2013 11:22 AM, WM wrote:
> On 20 Apr., 17:30, fom <fomJ...@nyms.net> wrote:
>> On 4/20/2013 3:16 AM, WM wrote:
>>
>>> Matheology § 255
>>
>>> Let S = (1), (1, 2), (1, 2, 3), ... be a sequence of all finite
>>> initial sets s_n = (1, 2, 3, ..., n) of natural numbers n.
>>
>>> Every natural number is in some term of S.
>>> U s_n = |N
>>> forall n exists i : n e s_i.
>>
>>> S is constructed by adding s_(i+1) after s_(i).
>>
>> Notice that WM is claiming that the sequence has
>> a recursive definition.
>
> You can call it also inductive.
No. The domain of a recursive definition
is an inductive set.
f(0)=(1)={{1},{1,1}}={{1}}
f(1)=(1,2)={{{{1}}},{{{1}},2}}
f(n)=(1,...,n+1)={{f(n-1)},{f(n-1),n+1}}
And, one can have a proof by induction whereby
the form of a recursive definition facilitates
the proof of the inductive step.
I am certain that one could find "inductive
definition" in the literature. However, it
confuses the relationships. One has inductive
sets, inductive proofs, and recursive definitions.
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